In this paper, we propose and analyze two stabilized mixed finite element methods for the dual-porosity-Stokes model, which couples the free flow region and microfracture-matrix system through four interface conditions on an interface. The first stabilized mixed finite element method is a coupled method in the traditional format. Based on the idea of partitioned time stepping, the four interface conditions, and the mass exchange terms in the dual-porosity model, the second stabilized mixed finite element method is decoupled in two levels and allows a noniterative splitting of the coupled problem into three subproblems. Due to their superior conservation properties and convenience of the computation of flux, mixed finite element methods have been widely developed for different types of subsurface flow problems in porous media. For the mixed finite element methods developed in this article, no Lagrange multiplier is used, but an interface stabilization term with a penalty parameter is added in the temporal discretization. This stabilization term ensures the numerical stability of both the coupled and decoupled schemes. The stability and the convergence analysis are carried out for both the coupled and decoupled schemes. Three numerical experiments are provided to demonstrate the accuracy, efficiency, and applicability of the proposed methods. KEYWORDS decoupled numerical methods, dual-porosity-Stokes model, horizontal wellbore, mixed finite elements, stabilization Int J Numer Methods Eng. 2019;120:803-833. wileyonlinelibrary.com/journal/nme 804 AL MAHBUB ET AL.model, It is not surprising that a great deal of effort has been devoted to develop appropriate numerical methods to solve the (Navier-)Stokes-Darcy fluid flow system, including coupled finite element methods, 17-21 domain decomposition methods, 22-31 Lagrange multiplier methods, 3,32,33 mortar finite element methods, 34,35 least-square methods, 36-38 partitioned time-stepping methods, 39,40 two-grid and multigrid methods, 41-44 discontinuous Galerkin finite element methods, 45-50 boundary integral methods, 51,52 and many others.  Although the traditional Stokes-Darcy model has been well studied, it has limitation to describe the heterogeneity of the porous medium which contains multiple porosities. It is worth to notice that the realistic naturally fractured reservoir consists of two coexisting and interacting medium, namely, tight matrix and microfractures. Furthermore, it is more important to characterize the intrinsic properties and accurately modeled the flow interactions between each medium. 59,60 The first multiporosity model was proposed by Barenblatt et al 61 for the naturally fractured reservoir in 1960 where the microfracture and matrix systems are formulated by individual but overlapping continua. Warren and Root 62 developed a homogeneous orthotropic dual-porosity model in 1963 based on the model proposed by Barenblatt. There are many applications for the dual-porosity model such as the geothermal system, hydrogeology, pe...
In this paper, a two-level finite element method for Oseen viscoelastic fluid flow obeying an Oldroyd-B type constitutive law is presented. With the newly proposed algorithm, solving a large system of the constitutive equations will not be much more complex than the solution of one linearized equation. The viscoelastic fluid flow constitutive equation consists of nonlinear terms, which are linearized by taking a known velocity b(x), and transforms into the Oseen viscoelastic fluid flow model. Since Oseen viscoelastic fluid flow is already linear, we use a two-level method with a new technique. The two-level approach is consistent and efficient to study the coupled system which contains nonlinear terms. In the first step, the solution on the coarse grid is derived, and the result is used to determine the solution on the fine mesh in the second step. The decoupling algorithm takes two steps to solve a linear system on the fine mesh. The stability of the algorithm is derived for the temporal discretization and obtains the desired error bound. Two numerical experiments are executed to show the accuracy of the theoretical analysis. The approximations of the stress tensor, velocity vector, and pressure field are P1-discontinuous, P2-continuous and P1-continuous finite elements respectively.
In this paper, we present a stabilized lowest equal-order mixed finite element (FE) method for the Oseen viscoelastic fluid flow obeying an Oldroyd-B type constitutive law. To approximate the velocity, pressure, and stress tensor, we choose lowest equal-order FE triples p1-p1-p1 dg respectively. It is well known that these elements don't satisfy the inf-sup (or LBB) condition. Owing to the violation of the essential stability condition, the system became unstable. To overcome this difficulty, a standard pressure stabilization term is added to the discrete variational formulation, which ensures the well-posedness of the FE scheme. The existences and uniqueness of the FE scheme are derived. The desired optimal error bound is obtained. Three numerical experiments are executed to illustrate the validity and efficiency of the numerical method. The stabilized method provides attractive computational advantages, such as simpler data structures, parameter-free, no calculations of higher-order derivatives, and fast solver in simulations.
This paper considers numerical methods for solving the viscous incompressible steady-state Stokes-Darcy problem that can be implemented by the use of existing surface water and groundwater codes. In the porous medium problem for subsurface flow, a mixed discretization, which describes the macroscopic properties of a filtration process and is vigorous with respect to the variations in the material data, is often advocated. However, the theory of mixed spacial discretizations to Stokes-Darcy problems is far less developed than non-mixed versions. We develop herein a new robust stabilized fully mixed discretization technique in the porous media region coupled with the fluid region via the physically appropriate coupling conditions on the interface. The method developed here does not use any Lagrange multiplier and introduces a stabilization term in the temporal discretization to ensure the stability of the finite element scheme. The well-posedness of the finite element scheme and its convergence analysis are also derived. Finally, the efficiency and accuracy of the numerical methods are illustrated by several testing examples.
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